Characterization of local optima of polynomial modulus over a disc

Abstract

Given a complex polynomial p(z) of degree n, we are interested in characterization and computation of local optima — maxima or minima — of |p| over a disc D of radius r centered at the origin, or its boundary, ∂D. From the maximum modulus principle (MMP), together with parametrization of ∂D, maximization reduces to a univariate optimization. However, reduction to univariate optimization does not guarantee efficiency in the computation of local optima, nor gives any specific characterization, nor provides a bound on the number of local optima. In this article, we utilize the geometric modulus principle, a stronger result than MMP, to prove novel characterizations of local optima over D or ∂D. On the one hand, we prove any local optimum that lies on ∂D is necessarily a fixed point of $F_{r}(z)=\pm r(p/p^{\prime })/|p/p^{\prime }|$ and also give a sufficiency condition. On the other hand, we show any fixed point is necessarily a root of a derived polynomialP_r(z) of degree 2n defined in terms of p, $p^{\prime }$ and their conjugate reciprocals. This also proves the number of local optima is at most 2n, an upper bound that can be attained. Our results imply that all desired local optima can be approximated to arbitrary precision in polynomial time. In particular, the classic $\| p \|_{\infty }$ can be computed efficiently. Implementation of our algorithm using Matlab and MPSolve is capable of computing all roots of P_r(z) of degree up to 250 and subsequently all local optima of |p| to precision of 10^− 13 in less than 3 s. In summary, the proposed theoretical algorithm is also practical and efficient. Additionally, based on the derived formulas we propose specialized algorithms and present sample visualization through polynomiography.